This function estimates the first six L-moments of the 3-parameter Student t distribution given the parameters (\(\xi\), \(\alpha\), \(\nu\)) from parst3. The L-moments in terms of the parameters are
$$\lambda_1 = \xi\mbox{,}$$
$$\lambda_2 = 2^{6-4\nu}\pi\alpha\nu^{1/2}\,\Gamma(2\nu-2)/[\Gamma(\frac{1}{2}\nu)]^4\mbox{\, and}$$
$$\tau_4 = \frac{15}{2} \frac{\Gamma(\nu)}{\Gamma(\frac{1}{2})\Gamma(\nu - \frac{1}{2})} \int_0^1 \! \frac{(1-x)^{\nu - 3/2}[I_x(\frac{1}{2},\frac{1}{2}\nu)]^2}{\sqrt{x}}\; \mathrm{d} x - \frac{3}{2}\mbox{,}$$
where \(I_x(\frac{1}{2}, \frac{1}{2}\nu)\) is the cumulative distribution function of the Beta distribution. The distribution is symmetrical so that \(\tau_r = 0\) for odd values of \(r: r \ge 3\).
The functional relation \(\tau_4(\nu)\) was solved numerically and a polynomial approximation made. The polynomial in turn with a root-solver is used to solve \(\nu(\tau_4)\) in parst3. The other two parameters are readily solved for when \(\nu\) is available. The polynomial based on \(\log{\tau_4}\) and \(\log{\nu}\) has nine coefficients with a residual standard error (in natural logarithm units of \(\tau_4\)) of 0.0001565 for 3250 degrees of freedom and an adjusted R-squared of 1. A polynomial approximation is used to estimate the \(\tau_6\) as a function of \(\tau_4\); the polynomial was based on the theoLmoms estimating \(\tau_4\) and \(\tau_6\). The \(\tau_6\) polynomial has nine coefficients with a residual standard error units of \(\tau_6\) of 1.791e-06 for 3593 degrees of freedom and an adjusted R-squared of 1.
lmomst3(para, bypoly=TRUE)An R
list is returned.
Vector of the L-moments. First element is \(\lambda_1\), second element is \(\lambda_2\), and so on.
Vector of the L-moment ratios. Second element is \(\tau\), third element is \(\tau_3\) and so on.
Level of symmetrical trimming used in the computation, which is 0.
Level of left-tail trimming used in the computation, which is NULL.
Level of right-tail trimming used in the computation, which is NULL.
An attribute identifying the computational source of the L-moments: “lmomst3”.
The parameters of the distribution.
A logical as to whether a polynomial approximation of \(\tau_4\) as a function of \(\nu\) will be used. The default is TRUE because this polynomial is used to reverse the estimate for \(\nu\) as a function of \(\tau_4\). A polynomial of \(\tau_6(\tau_4)\) is always used.
W.H. Asquith with A.R. Biessen
Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8.
parst3, cdfst3, pdfst3, quast3
lmomst3(vec2par(c(1124,12.123,10), type="st3"))
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